If all of your supporters think vaccines cause diseases, if you're elected you're going to need to satisfy them by acting on those beliefs.
I think, in the abstract, you're right - having any sort of health procedure mandated by the govt is extremely problematic. However hugely communicable diseases are a big caveat to that principle, and we have been doing something about that for a long long time.
Typhoid Mary
was forcibly detained because, over and over again, she kept spreading the disease. Several different times, bird flu was halted before it could get too widespread because of govt mandated culls. Smallpox was completely eradicated because of vaccine efforts.
Imagine if in any of those cases people said 'freedom' was primary and prevented coercion for vaccines. The death and injury toll would be much higher by today, including those of the young and the immunocompromised.